= Solution
Let $a=L/2$, so each <mirror> in the symmetric <confocal optical resonator> has radius $R=L=2a$. At <mirror> 1, an incident beam whose <beam waist>[waist] is at the centre has $q_{\rm in}=a+iz_R$, with distances measured along its incoming direction. Reflection gives
$$
q_{\rm refl}=\frac{q_{\rm in}}{1-q_{\rm in}/a}
=-a+i\frac{a^2}{z_R}.
$$
After propagating the distance $a$ back to the centre, its parameter is purely imaginary. Thus the reflected <beam waist>[waist] is also at the <optical cavity>[cavity] centre and
$$
\boxed{z_R'=\frac{a^2}{z_R},\qquad z_Rz_R'=\left(\frac L2\right)^2.}
$$
The incident and reflected <beam waist>[waist] radii need not agree. With axial position $z$ measured from the centre, their radii obey
$$
w^2(z)=\frac\lambda\pi\left(z_R+\frac{z^2}{z_R}\right),\qquad
w'^2(z)=\frac\lambda\pi\left(\frac{a^2}{z_R}+\frac{z^2z_R}{a^2}\right).
$$
At either <mirror>, $z^2=a^2$ and the two <Gaussian beam radius>[spot sizes] agree, as reflection requires. Between the <mirrors> they generally have different waists while still reproducing themselves after a full round trip.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2019-confocal-beam-envelopes.png]
{title=Incident and reflected Gaussian envelopes in a symmetric confocal resonator}
{description=Two unequal waist radii at the same central plane have reciprocal Rayleigh ranges $z_Rz_R'=(L/2)^2$. The envelopes meet at both mirrors. The matched fundamental mode is included for comparison.}
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